In
combinatorial game theory, the zero game is the game where neither player has any legal options. Therefore, under the
normal play convention, the first player automatically loses, and it is a second-player win. The zero game has a
Sprague–Grundy value of zero. The combinatorial notation of the zero game is: .
[.]
A zero game should be contrasted with the
star game , which is a first-player win since either player must (if first to move in the game) move to a zero game, and therefore win.
Examples
Simple examples of zero games include
Nim
Nim is a mathematical two player game.
Nim or NIM may also refer to:
* Nim (programming language)
* Nim Chimpsky, a signing chimpanzee Acronyms
* Network Installation Manager, an IBM framework
* Nuclear Instrumentation Module
* Negative index met ...
with no piles or a
Hackenbush
Hackenbush is a two-player game invented by mathematician John Horton Conway. It may be played on any configuration of colored line segments connected to one another by their endpoints and to a "ground" line.
Gameplay
The game starts with the p ...
diagram with nothing drawn on it.
Sprague-Grundy value
The
Sprague–Grundy theorem applies to
impartial games (in which each move may be played by either player) and asserts that every such game has an equivalent Sprague–Grundy value, a "nimber", which indicates the number of pieces in an equivalent position in the game of
nim
Nim is a mathematical two player game.
Nim or NIM may also refer to:
* Nim (programming language)
* Nim Chimpsky, a signing chimpanzee Acronyms
* Network Installation Manager, an IBM framework
* Nuclear Instrumentation Module
* Negative index met ...
. All second-player win games have a Sprague–Grundy value of zero, though they may not be the zero game.
For example, normal
Nim
Nim is a mathematical two player game.
Nim or NIM may also refer to:
* Nim (programming language)
* Nim Chimpsky, a signing chimpanzee Acronyms
* Network Installation Manager, an IBM framework
* Nuclear Instrumentation Module
* Negative index met ...
with two identical piles (of any size) is not the zero game, but has value 0, since it is a second-player winning situation whatever the first player plays.
It is not a
fuzzy game
In combinatorial game theory, a fuzzy game is a game which is ''incomparable'' with the zero game: it is not greater than 0, which would be a win for Left; nor less than 0 which would be a win for Right; nor equal to 0 which would be a win for the ...
because first player has no winning option.
[.]
References
{{reflist
Combinatorial game theory
0 (number)